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CHAPTER 71
ASSESSMENT OF THE RELIABILITY OF CONCRETE SLAB BRIDGES
P. Thoft-Christensen*, F.M. Jensen**, C. Middleton*** & A. Blackmore***
*Aalborg University, Aalborg, Denmark
** CSRconsult, Aalborg, Denmark
***Cambridge University, Cambridge, UK
ABSTRACT
This paper is based on research performed for the Highways Agency, London, UK under the project DPU/9/44 Revision of Bridge Assessment Rules Based on Whole Life Performance: Concrete Bridges. It contains details on a methodology which can be used to generate Whole Life (WL) reliability profiles.
The reliability analysis is based on two limit states: collapse and crack width. Deterioration is chloride induced corrosion of the reinforcement. The methodology is illustrated by an example where the reliability profile is estimated for an expected lifetime of 120 years. Finally the sensitivity analysis for both limit states is performed.
Limit States
Two limit states are selected for the reliability analysis:
an ultimate limit state (ULS): collapse limit state (using yield line analysis)
a serviceability limit state (SLS): crack width limit state (using linear elasticity analysis)
Collapse (yield line) limit state
The following safety margin is used
EMBED Equation.2 (1)
where V is a model uncertainty variable, EMBED Equation.2 is the energy dissipated in yield lines, and EMBED Equation.2 is the work done by the applied loads.
The plastic collapse analysis and estimation of the load capacity is determined using the COBRAS program, see Middleton [1]. The reliability analysis (element and system) is done using programs RELIAB01 [2] and RELIAB02 [3]. The programs RELIAB and COBRAS have been interfaced and include an optimisation algorithm to determine the optimal yield line pattern in each iteration in the reliability analysis. In Thoft-Christensen [4] it is shown that using a fixed deterministic yield line pattern in the reliability analysis may lead to erroneous results. The estimation of the deterioration of the steel reinforcement is based on the program CORROSION [5]. The basic variables used in the yield line ULS are: thickness of the slab, cube strength of the concrete, density of the concrete, depth of reinforcement, yield strength of reinforcement, and two load parameters.
Crack width limit state
Cracking shall be limited to a level that will not impair the proper functioning of the structure or cause its appearance to be unacceptable. In the absence of specific requirements (e.g. water tightness), it may be assumed that limitation of the maximum design crack width to about 0.3 mm will generally be satisfactory for reinforced concrete members with respect to appearance and durability. The design crack width may be obtained from, see EUROCODE2 [6]
EMBED Equation.DSMT4 (2)
where EMBED Equation is the design crack width, EMBED Equation is the average final spacing, EMBED Equation is the mean strain allowing under the relevant combination of loads for the effects of tension stiffening, shrinkage, etc., and EMBED Equation is a coefficient relating the average crack width to the design value. EMBED Equation = 1.7 for load induced cracking. EMBED Equation may be calculated from the relation
EMBED Equation (3)
where EMBED Equation is the stress in the reinforcement calculated on the basis of a cracked section, EMBED Equation is the stress in the reinforcement calculated on the basis of a cracked section under the loading conditions causing first cracking. EMBED Equation is a coefficient which takes account of the bond properties of the bars. It is = 1.0 for high bond bars, and = 0.5 for plain bars. EMBED Equation is a coefficient which takes account of the duration of the loading or of repeated loading. It is = 1.0 for single, short term loading, and = 1.5 for a sustained load or for many cycles of repeated loading.
The average final crack spacing (in mm) for members subjected dominantly to flexure or tension can be calculated from the equation
EMBED Equation (4)
where EMBED Equation is the bar size in use (or the average bar size). EMBED Equation is the effective reinforcement ratio, EMBED Equation , where EMBED Equation.2 is the area of reinforcement contained within the effective tension area, EMBED Equation . EMBED Equation is a coefficient which takes account of the bond properties of the bar. It is = 0.8 for high bond bars and = 1.6 for plain bond bars. EMBED Equation is a coefficient which takes account of the strain distribution. It is = 0.5 for bending and = 1.0 for pure tension.
The crack width limit state can then be formulated by
EMBED Equation (5)
where EMBED Equation is a model uncertainty stochastic variable.
The stochastic variables used in the crack SLS are: concrete cover, distance between reinforcement bars, diameter of reinforcement bars, thickness of slab, elastic modulus of reinforcement bars, tensile strength of concrete, external bending moment, and one model uncertainty variable.
Deterioration modelling
Several models can be used to model the deterioration of reinforcement steel in concrete slabs. However, there is a general agreement that the model presented below is acceptable in most cases. Corrosion initiation period refers to the time during which the passivation of steel is destroyed and the reinforcement starts to corrode actively. Practical experience shows that the initiation stage is completely dominated by the carbonation of the concrete cover zone, and the excessively high chloride content around the embedded steel.
The rate of chloride penetration into concrete, as a function of depth from the concrete surface and time, can be represented by Fick's law of diffusion as follows:
EMBED Equation (6)
where EMBED Equation is the chloride ion concentration, as % of the weight of cement, at distance EMBED Equation cm from the concrete surface after EMBED Equation seconds of exposure to chloride source. EMBED Equation is the chloride diffusion coefficient expressed in cm2/sec. The solution of the differential equation (6) is
EMBED Equation.DSMT4 (7)
where EMBED Equation is the equilibrium chloride concentration on the concrete surface, as % of the weight of cement, EMBED Equation is the distance from the concrete surface in cm, EMBED Equation is the time in sec, erf is the error function, EMBED Equation is the diffusion coefficient in cm2/sec and EMBED Equation is the chloride concentration at any position EMBED Equation at time EMBED Equation . In a real structure, if EMBED Equation is assumed to be the chloride corrosion threshold and EMBED Equation is the thickness of concrete cover, then the corrosion initiation period,EMBED Equation , can be calculated based on a knowledge of the parameters EMBED Equation and EMBED Equation . For bridge decks under de-icing conditions EMBED Equation =1.6, as % of cement weight, is often used.
The time EMBED Equation.2 to initiation of reinforcement corrosion is
EMBED Equation (8)
where EMBED Equation.2 is the initial chloride concentration, EMBED Equation.2 is the critical chloride concentration by which corrosion starts, and EMBED Equation.2 is the concrete cover. For plain concrete of moderate strength (EMBED Equation ) reported values of EMBED Equation are in the range between EMBED Equation and EMBED Equation .
When corrosion has started then the diameter EMBED Equation.2 of the reinforcement bars at time t is modelled by
EMBED Equation.2 (9)
where EMBED Equation.2 is the initial diameter, EMBED Equation.2 is a corrosion coefficient, and EMBED Equation.2 is the rate of corrosion. The area of a reinforcement bar is then modelled using the following formulation
EMBED Equation.DSMT4 (10)
EMBED Equation.2 , EMBED Equation.2
A(t) is the area of reinforcement bars EMBED Equation.2 at the time t years, n is the number of reinforcement bars, EMBED Equation.2 is the diameter of a single bar EMBED Equation.2 and EMBED Equation.2 is the corrosion initiation time in years. The value "0.0203" in the estimation of EMBED Equation.2 will vary depending on the circumstances.
The initiation time of corrosion is determined based on values of EMBED Equation.2 . After the deterioration is started the corrosion rate is modelled by the corrosion current EMBED Equation.2 only. The model for EMBED Equation.2 (and the used EMBED Equation.2 value) relates to an average deterioration of the reinforcement in the concrete. An important aspect of corrosion in addition to the average corrosion is the maximum penetration (pitting of reinforcement). Pitting of reinforcement may have more influence on the reliability than the average deterioration due to localised much higher weakening of the reinforcement. The ratio EMBED Equation.2 between the maximum penetration EMBED Equation.2 and the average penetration EMBED Equation.2 has been estimated by a number authors to be between 4-10, see e.g. Gonzсlez et al.[7]. Pitting corrosion is not included in this investigation.
The stochastic variables used in the deterioration modelling are: initial chloride concentration on surface, initial chloride concentration in concrete, diffusion coefficient for the concrete, cover on rebar, critical chloride concentration, and rate of corrosion.
Example
The following example is used to illustrate the proposed methodology. The example is based on an existing UK bridge, but some limitations and simplifications are made. The bridge was built in 1975.
The bridge is designed for 45 units HB load; see Department of Transport [8]. The bridge has a span of 9.755 m, the width is 2 EMBED Equation 13.71 m, and the slab thickness is 550 mm (see figure 1). Based on the corrosion data shown in table 1 the expected area of the reinforcement as function of time can be calculated, see figure 2.
EMBED Word.Picture.6
Figure 1. Bridge data.
Figure 2. Reinforcement area A(t) as function of time.
Reliability profiles for the two limit states used in this project are calculated on basis of the stochastic modelling shown in tables 1 and 2.
Stochastic variables: Yield line limit stateNoTypePar. 1Par. 2Description1Normal550.010.0Thickness of slab2Lognormal30.06.0Cube strength of concrete3Normal23.60.4Density of concrete4Lognormal289.025.0Yield strength: Longitudinal reinforcement5Normal60.08.0Cover on Longitudinal reinforcement6Lognormal289.025.0Yield strength: transverse reinforcement7Normal86.08.0Cover on transverse reinforcement8Fixed10053.0-Longitudinal reinforcement area9Fixed565.0-Transverse reinforcement area10Gumbel 0.3520.026Static load factor11Normal1.270.20Dynamic load factor12Normal1.080.072Chloride concentration on surface [%]13Fixed0.0-Initial chloride concentration [%]14Normal35.02.5Diffusion Coefficient 15Normal0.40.05Critical Chloride concentration16Uniform2.50.29Corrosion parameters17Normal1.00.05Model uncertainty variableTable 1. Stochastic modelling used for the ULS.
Stochastic variables: Crack width limit stateNoTypePar. 1Par. 2Description1Normal60.09.0Concrete cover [mm]2Normal125.012.5Distance between reinforcement bars [mm]3Normal40.01.2Diameter of reinforcement bar [mm]4Normal550.027.0Thickness of slab [mm]5Normal200.0E36.0E3Young's modulus [N/mm2]6Normal3.40.68Tensile strength [N/mm2]7Gumbel1.00.10Model uncertainty [-]8Gumbel0.3520.026Static loading factor [-]9Normal1.270.20Dynamic loading factor [-]10Normal1.080.072Chloride concentration on surface [%]11Fixed0.0-Initial chloride concentration [%]12Normal35.02.5Diffusion Coefficient 13Normal0.40.05Critical Chloride concentration14Uniform2.50.29Corrosion parametersTable 2. Stochastic modelling used for the SLS.
Figure 3. Reliability profiles using a yield line limit state.
The general traffic highway load model in the Eurocode 1, Part 3 (ENV 1991-3:1995) for lane and axle load is applied. The load effects produced by the Eurocode model (lane and axle load) are multiplied by a static loading factor (extreme type 1) and a dynamic loading factor (normal). See e.g. stochastic variables 10 and 11 used for the yield line limit state. Several load cases are considered in the project. In this paper only the load case with all packed lanes of 3 m width is included.
The normalised reliability profiles for the yield line ULS (full width failure) and the corresponding failure of probability profile are shown in figure 3. The reliability index at time t=0 is 11.4. Due to the size of the concrete cover (mean value 60 mm) the deterioration does not have any effect until year 70.
The results from the sensitivity analysis with regard to the mean values are shown for t =0 years and t =120 years in figure 4. The most important variables are as expected the thickness of the slab and the yield strength of the reinforcement. Observe that the magnitude of sensitivity with regard to the cover changes from negative at time t =0 years to positive at time t =120 years due to the corrosion.
Figure 4 : Sensitivity analysis for yield line limit state at t = 0 years
and at t = 120 years.
Figure 5 : Reliability profiles using a crack width limit state.
The normalised reliability profiles for the crack SLS and the corresponding failure of probability profile are shown in figure 5. The reliability index at time t =0 is 7.3. Due to the size of the concrete cover (mean value 60 mm) the deterioration does not have any effect until year 90.
The results from the sensitivity analysis with regard to the mean values are shown for t =0 years and t =120 years in figure 6. The most important variables are as expected the concrete cover, the diameter of the reinforcement, the thickness of the slab, and Young's modulus. Observe that the magnitude of the sensitivity with regard to the cover is decreasing from time t =0 years to time t =120 years due to the corrosion.
Figure 6 : Sensitivity analysis for yield line limit state at t = 0 years
and at t = 120 years.
From a bridge management point of view it is often relevant to define the service life time of a bridge as the corrosion initiation time. With this definition the service life time for the bridge in question is about 70 years.
Acknowledgement
The authors would like to thank the Highways Agency, London for permission to publish this paper. The work herein was carried out under a contract placed on CSRconsult by the HA. Any views expressed in this paper are not necessarily those of the Highways Agency or of the UK Department of Transport.
References
[1] Middleton, C. Example Collapse & Reliability Analyses of Concrete Bridges Using a new Analyses Technique. Cambridge University, UK, 1994.
[2] RELIAB01, Version 2.0, Manual and Software. CSR-software, Aalborg, Denmark, 1994.
[3] RELIAB02, Version 2.0, Manual and Software. CSR-software, Aalborg, Denmark, 1994.
[4] Thoft-Christensen, P. Reliability of Plastic Slabs. Proc. ICOSSAR Conference, San Francisco, USA, 1989.
[5] CORROSION. Version 1.0, Manual and Software. CSR-software, Aalborg, Denmark, 1995.
[6] Eurocode 2. Design of concrete structures - Part 1-1: General rules and rules for buildings, ENV 1992-1:1991.
[7] Gonzсlez, J.A., C. Andrade, C. Alonso & S. Feliu. Comparison of Rates of General Corrosion and Maximum Pitting Penetration on Concrete Embedded Steel Reinforcement. Cement and Concrete Research 25:2, 1995, pp. 257-264.
[8] Department of Transport. Loads for Highway Bridges. Departmental Standard BD37/88, London, 1989.
Proc. IFIP WG 7.5 Conf. Boulder, Colorado, USA, April 1996, pp. 321-328.
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